Ann. Funct. Anal. 1 (2010), no. 1, 72–79
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL: www.emis.de/journals/AFA/
ON n-NORMS AND BOUNDED n-LINEAR FUNCTIONALS IN
A HILBERT SPACE
S. M. GOZALI1 , H. GUNAWAN2∗ AND O. NESWAN3
Communicated by Y. Seo
Abstract. In this paper we discuss the concept of n-normed spaces. In particular, we show the equality of four different formulas of n-norms in a Hilbert
space. In addition, we study the notion of bounded n-linear functionals on an
n-normed space and present some results on it.
1. Introduction and preliminaries
Let X be a real vector space with dim(X) ≥ 2. A real-valued function k·, ·k :
X × X −→ R is called a 2-norm on X if the following conditions hold:
A1. kx, yk = 0 if and only if x, y are linearly dependent;
A2. kx, yk = ky, xk for all x, y ∈ X;
A3. kαx, yk = |α| kx, yk for all α ∈ R, x, y ∈ X;
A4. kx + y, zk ≤ kx, zk + ky, zk for all x, y, z ∈ X.
The pair (X, k·, ·k) is then called a 2-normed space.
The notion of 2-normed spaces has been extended to that of n-normed spaces.
So, from now on, let n be a nonnegative integer and X be a real vector space
of dimension d ≥ n. A real-valued function k·, . . . , ·k on X n = X × · · · × X (n
factors) satisfying the following four properties:
B1. kx1 , . . . , xn k = 0 if and only if x1 , . . . , xn are linearly dependent;
B2. kx1 , . . . , xn k is invariant under permutation for all x1 , . . . , xn ∈ X;
B3. kαx1 , x2 , . . . , xn k = |α| kx1 , x2 , . . . , xn k for all α ∈ R, x1 , . . . , xn ∈ X;
B4. kx0 +x1 , x2 , . . . , xn k ≤ kx0 , x2 , . . . , xn k+kx1 , x2 , . . . , xn k for all x0 , x1 , . . . ,
xn ∈ X,
Date: Received: 15 July 2010; Accepted: 19 October 2010.
∗
Corresponding author.
2010 Mathematics Subject Classification. Primary 46B20; Secondary 46B99, 46C05, 46C99.
Key words and phrases. n-normed space, bounded n-linear functional.
72
n-NORMS AND BOUNDED n-LINEAR FUNCTIONALS
73
is called an n-norm on X, and the pair (X, k·, . . . , ·k) is called an n-normed space.
If X is a normed space with dual X 0 , then — as formulated by Gähler [3] —
we may define an n-norm on X by
f1 (x1 ) · · · fn (x1 ) ..
..
...
=
sup
det [fj (xi )].
kx1 , . . . , xn kG :=
sup
.
.
fj ∈X 0 , kfj k≤1 fj ∈X 0 , kfj k≤1
f1 (xn ) · · · fn (xn )
Meanwhile, if X is equipped with an inner product h·, ·i, we can define the standard n-norm on X by
q
S
kx1 , . . . , xn k := det [hxi , xj i].
The value of kx1 , · · · , xn kS is nothing but the volume of the n-dimensional parallelepiped spanned by x1 , . . . , xn .
The concept of 2-normed spaces and, more generally, that of n-normed spaces,
were initially developed by Gähler [2, 3, 4, 5] in the 1960’s. Since then, many
researchers have developed and obtained various results, see e.g. [7, 8, 10, 11, 14].
Related works can be found, for examples, in [1, 9, 13].
Our interest here is to study various formulas of n-norms, especially in a Hilbert
space. So far, we already have two formulas of n-norms, Gähler’s formula and the
standard one. In the next section, we show that the two formulas are identical.
As it turns out, we also have two more definitions of n-norms, which have different
formulas but are actually the same with the previous two. Moreover, we study
the notion of bounded n-linear functionals on n-normed spaces and present some
facts, including that in the standard case.
2. Main results
2.1. n-norms in a Hilbert space. Hereafter, let X be a real Hilbert space,
unless otherwise stated. By Riesz Representation Theorem, each f ∈ X 0 can be
identified by an element y ∈ X such that f (x) = hx, yi for all x ∈ X. Hence
Gähler’s formula on X reduces to
kx1 , . . . , xn kG =
sup
det [hxi , yj i].
yj ∈X, kyj k≤1
As shown in [11], this formula is actually identical with the standard one. Indeed,
applying the generalized Cauchy-Schwarz inequality [12],
kx1 , . . . , xn kG ≤
sup
kx1 , . . . , xn kS ky1 , . . . , yn kS .
yj ∈X, kyj k≤1
Now, by Hadamard’s inequality [6],
ky1 , . . . , yn kS ≤ ky1 k · · · kyn k,
and so we get
kx1 , . . . , xn kG ≤ kx1 , . . . , xn kS .
74
S.M. GOZALI, H. GUNAWAN, O. NESWAN
Conversely, assuming that x1 , . . . , xn are linearly independent, let x01 , . . . , x0n be
the vectors obtained from x1 , . . . , xn through the Gram-Schmidt orthogonalization process. Then
kx1 , . . . , xn kS = kx01 k · · · kx0n k.
Now, if yj = kx10 k x0j , j = 1, . . . , n, then by the properties of determinants, we
j
have
0 0
hx1 , y1 i ... hx1 , yn i hx1 , x1 i ... hx01 , x0n i ..
..
..
..
...
...
= 0 1 0 .
.
.
.
kx1 k···kxn k hx , y i ... hx , y i hx0 , x0 i ... hx0 , x0 i n 1
n n
n
1
n
n
= kx01 k · · · kx0n k.
Hence we obtain
kx1 , . . . , xn kG ≥ kx1 , . . . , xn kS .
Therefore, the Gähler’s formula is identical with the standard one, as claimed.
Suppose that X is separable, and {e1 , e2 , . . . } is a complete orthonormal set in
X. Then, any member x of X may be identified by the sequence (hx, ej i), which
is in `2 . As in [7], we can define an n-norm on X by
"
#1/2
X
1 X
kx1 , . . . , xn k2 :=
···
|det [αijk ]|2
,
n! j
j
1
n
where αij := hxi , ej i. By Parseval’s identity, properties of determinants, and limiting arguments [7], this n-norm may be derived from the standard one. Therefore,
we have:
Proposition 2.1. On a separable Hilbert space, the three formulas of n-norms,
namely k·, . . . , ·kG , k·, . . . , ·kS and k·, . . . , ·k2 coincide.
In addition to the above formulas, we shall introduce another formula of n-norm
— which we state in the following proposition.
Proposition 2.2. The following function
kx1 , . . . , xn kD :=
sup
y1 , . . . , y n ∈ X
ky1 , . . . , yn kS ≤ 1
hx1 , y1 i · · · hx1 , yn i
..
..
..
.
.
.
hx , y i · · · hx , y i
n 1
n n
defines an n-norm on X.
Proof. It is obvious that, if x1 , . . . , xn are linearly dependent vectors, then we have
kx1 , . . . , xn kD = 0. Conversely, if kx1 , . . . , xn kD = 0, then the rows of the matrix
[hxi , yj i] are linearly dependent for all y1 , . . . , yn ∈ X with ky1 , . . . , yn kS ≤ 1.
This happens only if x1 , . . . , xn are linearly dependent.
Next, by the properties of determinants, we have the invariance of kx1 , . . . , xn kD
under permutation.
Furthermore, we have kαx1 , x2 , . . . , xn kD = |α| kx1 , x2 , . . . , xn kD for all α ∈ R.
n-NORMS AND BOUNDED n-LINEAR FUNCTIONALS
Finally, for arbitrary elements x0 , x1 , . . . , xn in X, we have
hx0 + x1 , y1 i · · · hx0 + x1 , yn i hx0 , y1 i · · · hx0 , yn i ..
..
..
..
..
..
=
.
.
.
.
.
.
hx , y i
hx , y i · · · hx , y i ···
hxn , yn i n 1
n 1
n n
hx1 , y1 i · · · hx1 , yn i
..
..
..
+ .
.
.
hx , y i · · · hx , y i
n 1
n n
75
.
Taking the supremums of both sides, we obtain
kx0 + x1 , . . . , xn kD ≤ kx0 , . . . , xn kD + kx1 , . . . , xn kD .
This completes the proof.
Regarding the last formula, we have the following proposition.
Proposition 2.3. The two formulas k·, . . . , ·kG and k·, . . . , ·kD are identical.
Proof. Let y1 , . . . , yn be elements in X. Since ky1 , . . . , yn kS ≤ ky1 k · · · kyn k, we
have ky1 , . . . , yn kS ≤ 1 whenever kyj k ≤ 1 for j = 1, . . . , n. It thus follows that
kx1 , . . . , xn kG ≤ kx1 , . . . , xn kD .
Conversely, if 0 < ky1 , . . . , yn kS ≤ 1, then by the generalized Cauchy-Schwarz
inequality we have
p
p
det [hxi , yj i] ≤ det [hxi , xj i] det [hyi , yj i]
.
= kx1 , . . . , xn kS ky1 , . . . , yn kS
S
G
≤ kx1 , . . . , xn k = kx1 , . . . , xn k
Hence we obtain
kx1 , . . . , xn kD ≤ kx1 , . . . , xn kG .
Therefore the two formulas are identical.
Corollary 2.4. On a separable real Hilbert space, the four formulas of n-norms,
k·, . . . , ·kG , k·, . . . , ·kS , k·, . . . , ·k2 , and k·, . . . , ·kD , are identical.
The last formula makes us realize that we can actually define an n-norm on
the dual of an n-normed space. Recall that we can define a norm on the dual X 0
of a normed space (X, k · k) by the formula
kf k := sup |f (x)|,
f ∈ X 0.
kxk≤1
Now, if X is equipped with an n-norm k·, . . . , ·k and X 0 denotes the dual of X,
we would like to induce an n-norm on X 0 from the n-norm k·, . . . , ·k on X. It
turns out that it is possible to do so, as it is shown in the following proposition.
The proof is similar to that of Proposition 2.2, and so we do not repeat it here.
76
S.M. GOZALI, H. GUNAWAN, O. NESWAN
Proposition 2.5. Let (X, k·, . . . , ·k) be an n-normed space. Then, the function
k·, . . . , ·k0 : (X 0 )n → R given by the formula
f1 (x1 ) . . . fn (x1 ) ..
..
..
kf1 , . . . , fn k0 :=
sup
.
.
.
xi ∈ X
f (x ) . . . f (x ) 1 n
n n
kx1 , . . . , xn k ≤ 1
defines an n-norm on X 0 .
Remark. Through Proposition 2.5 we have shown that the dual of an n-normed
space is also an n-normed space, whose n-norm is induced by the n-norm on the
original space.
2.2. Bounded n-Linear Functionals. In this part we shall observe the notion
of bounded n-linear functionals on n-normed spaces. Let (X, k·, . . . , ·k) be an
n-normed space. The function F : X n → R is called an n-linear functional on X
if F is linear in each variable. The n-linear functional F is bounded if there exists
k such that
|F (x1 , . . . , xn )| ≤ kkx1 , . . . , xn k,
(x1 , ..., xn ) ∈ X n .
If F is bounded, we define the norm of F by
|F (x1 , . . . , xn )|
,
kF k :=
sup
kx1 ,...,xn k6=0 kx1 , . . . , xn k
or equivalently
kF k :=
sup
|F (x1 , ..., xn )|.
kx1 ,...,xn k=1
The following proposition gives two alternative formulas for kF k.
Proposition 2.6. Let F be a bounded n-linear functional on X. Then, we have
kF k = inf{k : |F (x1 , . . . , xn )| ≤ kkx1 , . . . , xn k, (x1 , . . . , xn ) ∈ X n }
=
sup |F (x1 , . . . , xn )|.
kx1 ,...,xn k≤1
Proof. Let K = {k : |F (x1 , . . . , xn )| ≤ kkx1 , . . . , xn k, (x1 , . . . , xn ) ∈ X n }. It is
obvious that kF k ∈ K, and so inf K ≤ kF k. Conversely, for each k ∈ K, we have
|F (x1 , . . . , xn )|
≤k
kx1 , . . . , xn k
whenever kx1 , . . . , xn k =
6 0, so that kF k ≤ k. But since this is true for all k ∈ K,
we obtain kF k ≤ inf K. Therefore kF k = inf K.
Next, if kx1 , . . . , xn k ≤ 1, then
|F (x1 , . . . , xn )| ≤ kF k kx1 , . . . , xn k ≤ kF k.
This implies that
sup
|F (x1 , . . . , xn )| ≤ kF k.
kx1 ,...,xn k≤1
Conversely, we have
kF k =
sup
kx1 ,...,xn k=1
|F (x1 , . . . , xn )| ≤
sup
kx1 ,...,xn k≤1
|F (x1 , . . . , xn )|.
n-NORMS AND BOUNDED n-LINEAR FUNCTIONALS
Therefore, kF k =
|F (x1 , . . . , xn )|.
sup
77
kx1 ,...,xn k≤1
To give an example, consider the n-normed space (Rn , k·, . . . , ·kS ) with the
standard basis {e1 , . . . , en }. Define F by
α11 . . . α1n .. = det [α ]
..
F (x1 , . . . , xn ) := ...
.
ij
. α
.
.
.
α
n1
nn
Pn
where xi = j=1 αij ej , i = 1, . . . , n. Then, one may observe that F is a bounded
n-linear functional on Rn , with kF k = 1.
Moreover, we have the following result.
Proposition 2.7. Let (X, h·, ·i) be a real Hilbert space, which is also equipped
with the standard n-norm k·, . . . , ·kS . For fixed elements y1 , . . . , yn in X, define
F on X n by
F (x1 , . . . , xn ) = det [hxi , yj i].
Then F is a bounded n-linear functional on X and kF k = ky1 , . . . , yn kS .
Proof. Based on Proposition 2.6, the norm of F is given by
det [hxi , yj i].
kF k =
sup
kx1 ,...,xn kS ≤1
The generalized Cauchy-Schwarz inequality gives us
kF k ≤
kx1 , . . . , xn kS ky1 , . . . , yn kS ≤ ky1 , . . . , yn kS .
sup
S
kx1 ,...,xn k ≤1
Now, if we choose
yi
xi = p
n
ky1 , . . . , yn kS
,
then we see that
F (x1 , . . . , xn ) = ky1 , . . . , yn kS .
Therefore, we conclude that kF k = ky1 , . . . , yn kS .
Note that, on the space `2 (which is equipped with the n-norm k·, . . . , ·k2 ), the
above functional F in Proposition 2.7 may be rewritten by the formula
X
1 X
F (x1 , . . . , xn ) :=
···
det [xijk ] det [yijk ]
n! j
j
n
1
2
where y1 , . . . , yn are fixed in ` .
This formula also defines a bounded n-linear functional F on `p (1 ≤ p < ∞),
which is equipped with the n-norm k·, . . . , ·kp , where
"
# p1
X
X
1
kx1 , . . . , xn kp :=
···
|det [xijk ]|p
n! j
j
1
n
78
S.M. GOZALI, H. GUNAWAN, O. NESWAN
(see [7]). By Hölder’s inequality, we have
X
1 X
···
det [xijk ] det [yijk ] ≤ kx1 , . . . , xn kp ky1 , . . . , yn kq
n! j
j
1
for
1
p
+
1
q
n
= 1. Hence it follows that F is bounded, with kF k ≤ ky1 , . . . , yn kq .
2.3. Concluding Remark. We have introduced a new formula of n-norm in
a real Hilbert space which can be viewed as a modification of Gähler’s formula.
Accordingly, in a separable Hilbert space, we have four formulas of n-norms which
are identical. In addition, we have presented some results regarding the bounded
n-linear functionals on n-normed spaces, especially on Hilbert spaces (equipped
with the standard n-norm).
In connection with the theory of n-normed spaces, we have introduced a formula for the induced n-norm on the dual space of an n-normed space. The formula is defined in a natural way which is similar to that in the theory of normed
spaces. As a consequence of our result, we know that if X is an n-normed space,
then the dual X 0 is also an n-normed space. It should be interesting to study
the relation between an n-normed space and its dual, which is also an n-normed
space. In particular, one might inquire whether the following Riesz type representation theorem holds: Let (X, h·, ·i) be a separable real Hilbert space, which
is also equipped with the standard n-norm k·, . . . , ·kS . If F is a bounded n-linear
functional on X, then there exists unique fixed elements y1 , . . . , yn ∈ X such that
F (x1 , . . . , xn ) = det[hxi , yj i] for all x1 , . . . , xn ∈ X. The research on this problem
is still ongoing at the present time.
Acknowledgement. This Research is supported by the Directorate General of
Higher Education, Republic of Indonesia, through Fundamental Research Grant
2010.
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n-NORMS AND BOUNDED n-LINEAR FUNCTIONALS
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1
Department of Mathematics, Bandung Institute of Technology, Bandung,
Indonesia;
Permanent Address: Department of Mathematics Education, Indonesia University of Education, Bandung, Indonesia.
E-mail address: sumanang@students.itb.ac.id
2,3
Department of Mathematics, Bandung Institute of Technology, Bandung,
Indonesia;
Analysis and Geometry Research Group, ITB.
E-mail address: hgunawan@math.itb.ac.id, oneswan@math.itb.ac.id